#7224200

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,7,4,6)(11,15)(14,16)(17,18,23,28)(19,24,31,42)(20,26,34,44)(21,27,22,37)(25,36,45,33)(29,39,32,46)(30,40,35,41)(38,47,43,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,24)(20,32)(21,35)(22,30)(23,31)(25,43)(26,46)(27,41)(28,42)(29,34)(33,47)(36,48)(37,40)(38,45)(39,44);(1,3,5,4)(2,6,8,7)(10,13)(14,16)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,38)(26,37)(27,44)(28,45)(30,47)(31,32)(35,48)(39,40)(41,46)(42,43);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,29)(21,36)(22,33)(23,34)(24,39)(25,37)(27,45)(28,44)(30,48)(31,32)(35,47)(38,40)(41,43)(42,46);(1,4,5,3)(2,7,8,6)(10,13)(14,16)(17,21,23,22)(18,27,28,37)(19,30,31,35)(20,33,34,36)(24,40,42,41)(25,44,45,26)(29,47,32,48)(38,39,43,46)

Polycyclic presentation:

2^9|a^2=fi,c^2=f,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7131396);